Reznik's Identity

For $P$ and $Q$ Polynomials in $n$ variables,

$${\vert P\cdot Q\vert _2}^2 = \sum_{i_1,\ldots,i_n\geq 0}{{\v... ...\ldots,D_n)Q(x_1,\ldots,x_n)\vert _2}^2\over i_1!\cdots i_n!},$$ (1)

where $D_i\equiv\partial/\partial x_i$, $\vert X\vert _2$ is the Bombieri Norm, and

$$P^{(i_1,\ldots,i_n)}=D_1^{i_1}\cdots D_n^{i_n} P.$$

Bombieri's Inequality follows from this identity.

See also Beauzamy and Dégot's Identity